A new proof of Sárközy's theorem
Author:
Neil Lyall
Journal:
Proc. Amer. Math. Soc. 141 (2013), 22532264
MSC (2010):
Primary 11B30
Published electronically:
March 14, 2013
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Abstract: It is a striking and elegant fact (proved independently by Furstenberg and Sárközy) that in any subset of the natural numbers of positive upper density there necessarily exist two distinct elements whose difference is given by a perfect square. In this article we present a new and simple proof of this result by adapting an argument originally developed by Croot and Sisask to give a new proof of Roth's theorem.
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E. CROOT AND O. SISASK, A new proof of Roth's theorem on arithmetic progressions, Proc. Amer. Math. Soc. 137 (2009), no. 3, 805809. MR 2457417 (2010f:11018)
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M. HAMEL, N. LYALL, AND A. RICE, Improved bounds on Sárközy's theorem for quadratic polynomials, to appear in Int. Math. Res. Not. 2012, doi:10.1093/imrn/rns106.
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N. LYALL AND ´A. MAGYAR, Polynomial configurations in difference sets, J. Num. Theory 129/2 (2009), 439450. MR 2473891 (2009j:05029)
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N. LYALL AND ´A. MAGYAR, Polynomial configurations in difference sets (Revised version), arxiv.org/abs/0903.4504.
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N. LYALL AND ´A. MAGYAR, Sárközy's Theorem, www.math.uga.edu/lyall/Research/Sarkozy.pdf.
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H. L. MONTGOMERY, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conference Series in Mathematics, 84, Amer. Math. Soc., Providence, RI, 1994. MR 1297543 (96i:11002)
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J. PINTZ, W. L. STEIGER, AND E. SZEMERÉDI, On sets of natural numbers whose difference set contains no squares, J. London Math. Soc. 37 (1988), 219231. MR 928519 (89g:11019)
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I. Z. RUZSA, Difference sets without squares, Period. Math. Hungar. 15 (1984), 205209. MR 756185 (85j:11022)
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 1.
 A. BALOG, J. PELIKÁN, J. PINTZ, AND E. SZEMERÉDI, Difference sets without th powers, Acta Math. Hungar. 65 (1994), 165187. MR 1278767 (95d:11130)
 2.
 E. CROOT AND O. SISASK, A new proof of Roth's theorem on arithmetic progressions, Proc. Amer. Math. Soc. 137 (2009), no. 3, 805809. MR 2457417 (2010f:11018)
 3.
 H. FURSTENBERG, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. d'Analyse Math. 31 (1977), 204256. MR 0498471 (58:16583)
 4.
 B. GREEN, On arithmetic structures in dense sets of integers, Duke Math. J. 114 (2002), no. 2, 215238. MR 1920188 (2003i:11021)
 5.
 W. T. GOWERS, Additive and Combinatorial Number Theory, www.dpmms.cam.ac.uk/wtg10/addnoth.notes.dvi
 6.
 M. HAMEL AND I. ŁABA, Arithmetic structures in random sets, Integers: Electronic Journal of Combinatorial Number Theory 8 (2008), #4. MR 2373088 (2009a:11055)
 7.
 M. HAMEL, N. LYALL, AND A. RICE, Improved bounds on Sárközy's theorem for quadratic polynomials, to appear in Int. Math. Res. Not. 2012, doi:10.1093/imrn/rns106.
 8.
 T. KAMAE AND M. MENDÈS FRANCE, van der Corput's difference theorem, Israel J. Math. 31 (1978), no. 34, 335342. MR 516154 (80a:10070)
 9.
 J. LUCIER, Intersective sets given by a polynomial, Acta Arith. 123 (2006), no. 1, 5795. MR 2232502 (2007b:11155)
 10.
 N. LYALL, A simple proof of Sárközy's theorem, arxiv.org/abs/1107.0243.
 11.
 N. LYALL AND ´A. MAGYAR, Polynomial configurations in difference sets, J. Num. Theory 129/2 (2009), 439450. MR 2473891 (2009j:05029)
 12.
 N. LYALL AND ´A. MAGYAR, Polynomial configurations in difference sets (Revised version), arxiv.org/abs/0903.4504.
 13.
 N. LYALL AND ´A. MAGYAR, Sárközy's Theorem, www.math.uga.edu/lyall/Research/Sarkozy.pdf.
 14.
 H. L. MONTGOMERY, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conference Series in Mathematics, 84, Amer. Math. Soc., Providence, RI, 1994. MR 1297543 (96i:11002)
 15.
 J. PINTZ, W. L. STEIGER, AND E. SZEMERÉDI, On sets of natural numbers whose difference set contains no squares, J. London Math. Soc. 37 (1988), 219231. MR 928519 (89g:11019)
 16.
 I. Z. RUZSA, Difference sets without squares, Period. Math. Hungar. 15 (1984), 205209. MR 756185 (85j:11022)
 17.
 A. S´ARKÖZY, On difference sets of sequences of integers. III, Acta Math. Acad. Sci. Hungar. 31 (1978), 355386. MR 487031 (80j:10062b)
 18.
 P. VARNAVIDES, On certain sets of positive density, Journal London Math. Soc. 34 (1959), 358360. MR 0106865 (21:5595)
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Additional Information
Neil Lyall
Affiliation:
Department of Mathematics, The University of Georgia, Athens, Georgia 30602
Email:
lyall@math.uga.edu
DOI:
http://dx.doi.org/10.1090/S00029939201311628X
PII:
S 00029939(2013)11628X
Received by editor(s):
October 18, 2011
Published electronically:
March 14, 2013
Dedicated:
Dedicated to Steve Wainger on the occasion of his retirement
Communicated by:
Michael T. Lacey
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
